Three-dimensional Stable Matching with Cyclic Preferences

TL;DR

This paper proves the existence of stable matchings in three-dimensional cyclic preference problems with five agents per group, extending previous results for smaller group sizes and showing multiple solutions.

Contribution

It demonstrates that stable matchings exist for groups of five agents and identifies multiple stable matchings, advancing understanding of the problem's solvability.

Findings

Stable matching exists for groups of five agents.

At least two distinct stable matchings are found for five agents.

Previous results confirmed existence for groups of three and four.

Abstract

We consider the three-dimensional stable matching problem with cyclic preferences, a problem originally proposed by Knuth. Despite extensive study of the problem by experts from different areas, the question of whether every instance of this problem admits a stable matching remains unanswered. In 2004, Boros, Gurvich, Jaslar and Krasner showed that a stable matching always exists when the number of agents in each of the groups is three. In 2006, Eriksson, Sj\"ostrand and Strimling showed that a stable matching exists also when the number of agents in each group is four. In this paper, we demonstrate that a stable matching exists when each group has five agents. Furthermore, we show that there are at least two distinct stable matchings in that setting.